Question: The sum of two numbers is $78$, and their difference is $38$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 78}$ ${x-y = 38}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 116 $ $ x = \dfrac{116}{2} $ ${x = 58}$ Now that you know ${x = 58}$ , plug it back into $ {x+y = 78}$ to find $y$ ${(58)}{ + y = 78}$ ${y = 20}$ You can also plug ${x = 58}$ into $ {x-y = 38}$ and get the same answer for $y$ ${(58)}{ - y = 38}$ ${y = 20}$ Therefore, the larger number is $58$, and the smaller number is $20$.